Fact 2 
Let \(\mathbf {B}\in \mathit {\mathbb {R}}^n\)  be a nonsingular matrix. Then, \(\mathbf {B}^{-1}\)  is upper (lower) triangular if and only if \(\mathbf {B}\)  is upper (lower) triangular (Fact 3.20.5. in [1]}).

